Mention of Srimati Radharani in various Puranas, Veda

Srimati Radharani is mentioned in many Puranas like the Padma, Skanda, Narada, Brahmanda, Brahmavaivarta, Matsya, Shiva Puranas as well as the Devi Bhagavatam.

In the Padma Purana, her worship is mentioned in the 7th as well as 20th Chapters of the Bhumikanda (Volume 5 of Translation), in The Radhashtami and Radhadamodara Vrata.

Further, the Patalakhanda (Volume 6), Chapter 71 is about the greatness of Radhakrishna, and the subsequent few chapters describe the various Leelas of the divine couple, including one Leela where the Pandava Arjuna Is transported to Vrindavana, where he becomes a Gopika called Arjuni and is blessed with the darshan of Radhakrishna.

In the Narada Purana, (Volume 3), the 89th chapter describes the Yugala Sahasranama or 1000 names of both Radha and Krishna, where the first 500 names are Krishna's and the second 500 are Radharani’s names. It also includes, in its Uttarabhaga, a chapter about Vrindavana Mahatmya (Volume 5, Chapter 80), where Sage Narada becomes a gopi in Vrindavana to take part in Srimati Radharani’s Leelas.

In the Skanda Purana, Vaishnava Khanda, Bhagavata Mahatmya talks about Srimati Radharani. The 1st chapter (Volume 6) contains the famous statement, “Atma Tu Radhika Tasya” – Radharani is Krishna's soul. The Vasudeva Mahatmya (Volume 7), Chapters 16 and 17 describe Goloka and the manifestation of Radhakrishna for Narada.

In the Brahmanda Purana, Upodghata Parva, the last few chapters tell the story of Parashurama. In these chapters, Parashurama receives the hymn called Krishna Prema-Amrta from Rishi Agastya. He goes to Kailasa to meet Lord Shiva, where he chops off Lord Ganesha’s tusk. When Parvati Devi is furious with Parashurama, Lord Krishna and Radharani arrive from Goloka to appease Parvati. Their arrival and subsequent stotra is described in the 42nd and 43rd chapters of the Upodghata Parva (Volume 2 of Brahmanda Purana). The Stotra ‘Grhe Radha Vane Radha’ is also said to be from this Purana.

The Matsya Purana, when listing the goddesses of various cities, says, “Rukmini Dvaravatyam Tu Radha Vrindavane Vane.”

In the Shiva Purana, the Yuddhakanda of Rudrasamhita contains the story of Shankhachuda, who is actually Sridama, the Sakha of Lord Krishna in Goloka, who incarnates as a demon due to the curse of Radharani. In this, the 30th and 31st Chapter narrate how Lord Krishna and Radha (owing to this being a Shiva-centric Purana), plead forgiveness from Lord Shiva.

The Brahmavaivarta Purana is completely about Radhakrishna, though owing to a considerable number of interpolations in this Purana, and it's being well known, I leave it unquoted.

The Devi Bhagavata Purana dedicates its entire Navama Skandha (Ninth Canto) to the glories of Radhakrishna and their Leelas in Goloka and the rest of the universe.

Coming to Srimad Bhagavatam, there are many who say that Shukabrahma, who recited the Bhagavatam, was a pure devotee of Radhakrishna, and so uttering their pastimes would cause him to lose his consciousness and therefore leave the Purana incomplete. Hence he refers to her using common nouns such as ‘Kaachit’ (One Girl), ‘Saa Vadhuh’ (That Bride (of Krishna)), and ‘Priya’, (Beloved).

Apart from these references, the sloka Anayaradhito Nunam gives her name in the phrase Aaradhito - as purported by Sri Vishvanatha Chakravarti.

It is Radharani who delivers the famous 'Bhramara Gita' where she talks to a bee thinking it to be Krishna's messenger when Uddhava visits Vrindavana.

Also, the Second Canto, Fourth Chapter, when Shukabrahma Rishi says the Dhyana Slokas, he meditates upon Lord Hari in his Eternal Abode with all his Shaktis and Opulences. The name he gives to Vishnu’s Energy is ‘Raadhas’.

Apart from that, as user @brahmajijnasa said, the Brihad Brahma Samhita, Narada Pancaratra, Sanatkumara-samhita, Garga Samhita, Brihad-gautamiya-tantra, Urdhvamnaya-tantra and the Purusa-bodhini Upanishad give references to Srimati Radharani.

The Narada Pancharatra gives the Radha Kavacha and the Radha Sahasranama Stotra as told by Lord Shiva to Devi Parvati.

The Garga Samhita is also devoted to the story of Radhakrishna, and how the divine couple performed their Leelas in Vrindavana, and also describes their separation and eventual reunion in Siddhashrama, after the elapse of the hundred-year curse of separation.

In the Vishnu Purana, during the description of Rasa Leela in Chapter 13 of Book 5, there are seven slokas (slokas 32-38) describing Krishna leaving the Rasa Mandala and going off with a special Gopi into the forest. The other Gopis also describe their footprints, and Krishna's adorning her with flowers and leaving her some distance away. This is almost the same as in the Bhagavatam. The word used here (where there it is Anayaradhito) is Vishnur Abhyarchito Yaya - Vishnu was worshipped extensively by her. Therefore we can infer that this Gopika is Srimati Radharani.